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Year 2022, Volume: 51 Issue: 2, 421 - 429, 01.04.2022
https://doi.org/10.15672/hujms.784819

Abstract

References

  • [1] P. Balazs, J.P. Antoine and A. Grybos, Weighted and Controlled Frames, Int. J. Wavelets Multiresolut. Inf. Process. 8 (1), 109-132, 2010.
  • [2] I. Bogdanova, P. Vandergheynst, J.-P. Antoine, L. Jacques and M. Morvidone, Stere- ographic wavelet frames on the sphere, Applied Comput. Harmon. Anal. 19, 223-252, 2005.
  • [3] P. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3, 543-557, 1997.
  • [4] O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123, 2199-2202, 1995.
  • [5] O. Christensen, An introduction to Frame and Riesz Bases, Birkhäuser, Boston, 2003.
  • [6] O. Christensen and R.S. Laugesen, Approximately dual frames in Hilbert spaces and applications to Gabor frames, Sampl. Theory Sig. Image Process. 9 (3), 7789, 2011.
  • [7] M.A. Dehgan and M.A. Hasankhani, g-dual frames in Hilbert spaces, U.P.B. Sci. Bull. 75, 129-140, 2013.
  • [8] R. Duffin and A. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc. 72, 341-366, 1952.
  • [9] S.M. Ramezani, G-duals of continuous frames and their perturbations in Hilbert spaces, U.P.B. Sci. Bull. 82, 75-82, 2020.
  • [10] S.M. Ramezani and A. Nazari, g-orthonormal bases, g-Riesz bases and g-dual of g- frames, U.P.B. Sci. Bull. 78, 91-98, 2016.
  • [11] M. Rashidi-Kouchi, A. Rahimi and F.A. Shah, Duals and multipliers of controlled frames in Hilbert spaces, Int. J. Wavelets, Multiresolut. Inf. Process. 16 (5), 1-13, 2018.
  • [12] W. Sun, Stability of g-frames, J. Math. Anal. Appl. 326 (2), 858-868, 2007.

Controlled $g$-dual frames and their approximates in Hilbert spaces

Year 2022, Volume: 51 Issue: 2, 421 - 429, 01.04.2022
https://doi.org/10.15672/hujms.784819

Abstract

In this paper, we introduced and characterized the controlled $g$-duals of a frame in a separable Hilbert space $\mathcal{H}$ . Afterwards, we obtained new $C$-controlled $g$-dual frames from the given $C$-controlled $g$-dual frames. In addition, the approximation for controlled $g$-dual frames was defined and some of their properties were investigated. Finally, we characterized the relationship between approximately $C$-controlled dual and $C$-controlled $g$-dual.

References

  • [1] P. Balazs, J.P. Antoine and A. Grybos, Weighted and Controlled Frames, Int. J. Wavelets Multiresolut. Inf. Process. 8 (1), 109-132, 2010.
  • [2] I. Bogdanova, P. Vandergheynst, J.-P. Antoine, L. Jacques and M. Morvidone, Stere- ographic wavelet frames on the sphere, Applied Comput. Harmon. Anal. 19, 223-252, 2005.
  • [3] P. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3, 543-557, 1997.
  • [4] O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123, 2199-2202, 1995.
  • [5] O. Christensen, An introduction to Frame and Riesz Bases, Birkhäuser, Boston, 2003.
  • [6] O. Christensen and R.S. Laugesen, Approximately dual frames in Hilbert spaces and applications to Gabor frames, Sampl. Theory Sig. Image Process. 9 (3), 7789, 2011.
  • [7] M.A. Dehgan and M.A. Hasankhani, g-dual frames in Hilbert spaces, U.P.B. Sci. Bull. 75, 129-140, 2013.
  • [8] R. Duffin and A. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc. 72, 341-366, 1952.
  • [9] S.M. Ramezani, G-duals of continuous frames and their perturbations in Hilbert spaces, U.P.B. Sci. Bull. 82, 75-82, 2020.
  • [10] S.M. Ramezani and A. Nazari, g-orthonormal bases, g-Riesz bases and g-dual of g- frames, U.P.B. Sci. Bull. 78, 91-98, 2016.
  • [11] M. Rashidi-Kouchi, A. Rahimi and F.A. Shah, Duals and multipliers of controlled frames in Hilbert spaces, Int. J. Wavelets, Multiresolut. Inf. Process. 16 (5), 1-13, 2018.
  • [12] W. Sun, Stability of g-frames, J. Math. Anal. Appl. 326 (2), 858-868, 2007.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Sayyed Mehrab Ramezani 0000-0001-8039-5847

Publication Date April 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 2

Cite

APA Ramezani, S. M. (2022). Controlled $g$-dual frames and their approximates in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics, 51(2), 421-429. https://doi.org/10.15672/hujms.784819
AMA Ramezani SM. Controlled $g$-dual frames and their approximates in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics. April 2022;51(2):421-429. doi:10.15672/hujms.784819
Chicago Ramezani, Sayyed Mehrab. “Controlled $g$-Dual Frames and Their Approximates in Hilbert Spaces”. Hacettepe Journal of Mathematics and Statistics 51, no. 2 (April 2022): 421-29. https://doi.org/10.15672/hujms.784819.
EndNote Ramezani SM (April 1, 2022) Controlled $g$-dual frames and their approximates in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics 51 2 421–429.
IEEE S. M. Ramezani, “Controlled $g$-dual frames and their approximates in Hilbert spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, pp. 421–429, 2022, doi: 10.15672/hujms.784819.
ISNAD Ramezani, Sayyed Mehrab. “Controlled $g$-Dual Frames and Their Approximates in Hilbert Spaces”. Hacettepe Journal of Mathematics and Statistics 51/2 (April 2022), 421-429. https://doi.org/10.15672/hujms.784819.
JAMA Ramezani SM. Controlled $g$-dual frames and their approximates in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics. 2022;51:421–429.
MLA Ramezani, Sayyed Mehrab. “Controlled $g$-Dual Frames and Their Approximates in Hilbert Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, 2022, pp. 421-9, doi:10.15672/hujms.784819.
Vancouver Ramezani SM. Controlled $g$-dual frames and their approximates in Hilbert spaces. Hacettepe Journal of Mathematics and Statistics. 2022;51(2):421-9.